1. Field
This disclosure is generally related to wireless communications, and more particularly, to techniques for rotating and transmitting multidimensional constellations in wireless communication systems.
2. Background
In high data rate orthogonal frequency division multiplexing (OFDM) systems such as, but not limited to, WiMedia ultra-wideband (UWB), channel diversity can be a major concern. In such systems, related codewords may span multiple subcarriers. The distance between a first codeword and a second related codeword may not span enough of the subcarriers to maintain efficient communications. If some of the subcarriers fade, the codewords may be difficult to distinguish.
Various techniques have been employed to increase codeword diversity without providing a complete solution. One technique, discussed by Boutros and Viterbo in “Signal space diversity: A power- and bandwidth-efficient diversity technique for the Rayleigh fading channel,” IEEE Trans. Inform. Theory, vol. 42, pp. 502-518, March 1996; and M. L. McCloud, in “Analysis and design of short block OFDM spreading matrices for use on multipath fading channels,” IEEE Trans. Comm., vol. 53, pp. 656-665, April 2005, is to group several uncorrelated subcarriers to form a multidimensional constellation. In such systems, the subcarriers may be modeled as, or viewed as, the axes of the constellation.
However, any subcarrier/axes of the multidimensional constellation that collapse or fade may lead to constellation points falling on top of each other along the non-fading axes. The constellation points may no longer be discernible and the error rate in the transmission may increase. On the other hand, if the multidimensional constellation is properly rotated, the constellation points may remain distinct when one subcarrier collapses. If the multidimensional constellation is properly rotated the constellation may improve codeword diversity without an increase in transmitted power or bandwidth.
Known techniques for rotating multidimensional constellations include: (1) For space time codes (STC), minimizing the bit error rate (BER) as a function of a rotation vector, expressed as Θ, while expressing the BER as a weighted average; (2) Minimizing the symbol error rate (SER); and (3) minimizing the maximum of a modified Chernoff approximation for all vectors. Techniques for rotating multidimensional constellations may include the assumption that dimensions are complex.
Rotating multidimensional constellations for STCs by minimizing the BER as a function of the rotation vector Θ while expressing the BER as a weighted average is discussed by M. Brehler and M. K. Varanasi in “Training-codes for the noncoherent multi-antenna block-Rayleigh-fading channel,” Proc. 37th Conf. Information Sciences and Systems, Baltimore, Md., March 1214, 2003. This technique may work in the presence of coding where soft bits enter the succeeding stages in the receiver.
Rotating multidimensional constellations by minimizing the SER is discussed by M. L. McCloud in “Analysis and design of short block OFDM spreading matrices for use on multipath fading channels,” IEEE Trans. Comm., vol. 53, pp. 656-665, April 2005, which is entirely incorporated herein. Such techniques may work for uncoded systems with BER when the system does not include packet error rate (PER) requirements.
Rotating multidimensional constellations by minimizing the maximum of a modified Chernoff approximation for all vectors is discussed by Boutros and Viterbo, in the article referenced above. This technique may not improve the average error rate, and the technique may leave some codewords with little diversity, and therefore prone to errors when subcarrier fades occur. In a variation of this technique, the diversity order may be maximized to combat fading, which may result in a majority of error vectors spanning multiple dimensions. The variation may be suitable for uncoded modulations with PER, rather than BER, requirements. This variation may guarantee that the weakest (worst case) codeword has enough protection.
In their article referenced above, Boutros and Viterbo have also discussed techniques for rotating multidimensional constellations that may also include the use of a maximum product distance rule that may necessitate a full diversity for all error vectors. Such techniques may also include using a unique variable, expressed as “λ” by J. Boutros and E. Viterbo, in the article referenced above, in order to avoid dealing with the large number of possible rotations in high dimensions and the large number of variables associated with high dimensions. Techniques associated with the unique variable also include incorporating one or more constraints on the allowed rotations. Techniques also include the use of a Hadamard-like structure in which sub-matrices and their combinations are optimized.